Matrix-Based Multigrid: Theory and Applications
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Boston, MA
Springer US
2003
|
| Schriftenreihe: | Numerical Methods and Algorithms
2 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | Many important problems in applied science and engineering, such as the NavierStokes equations in fluid dynamics, the primitive equations in global climate modeling, the strain-stress equations in mechanics, the neutron diffusion equations in nuclear engineering, and MRIICT medical simulations, involve complicated systems of nonlinear partial differential equations. When discretized, such problems produce extremely large, nonlinear systems of equations, whose numerical solution is prohibitively costly in terms of time and storage. High-performance (parallel) computers and efficient (parallelizable) algorithms are clearly necessary. Three classical approaches to the solution of such systems are: Newton's method, Preconditioned Conjugate Gradients (and related Krylov-space acceleration techniques), and multigrid methods. The first two approaches require the solution of large sparse linear systems at every iteration, which are themselves often solved by multigrid methods. Developing robust and efficient multigrid algorithms is thus of great importance. The original multigrid algorithm was developed for the Poisson equation in a square, discretized by finite differences on a uniform grid. For this model problem, multigrid exhibits extremely rapid convergence, and actually solves the problem in the minimal possible time. The original algorithm uses rediscretization of the partial differential equation (POE) on each grid in the hierarchy of coarse grids that are used. However, this approach would not work for more complicated problems, such as problems on complicated domains and nonuniform grids, problems with variable coefficients, and non symmetric and indefinite equations. In these cases, matrix-based multi grid methods are in order |
| Beschreibung: | 1 Online-Ressource (XVI, 221 p) |
| ISBN: | 9781475737264 9781475737288 |
| ISSN: | 1571-5698 |
| DOI: | 10.1007/978-1-4757-3726-4 |
Internformat
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Datensatz im Suchindex
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| any_adam_object | |
| author | Shapira, Yair |
| author_facet | Shapira, Yair |
| author_role | aut |
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| author_variant | y s ys |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
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| dewey-search | 518 |
| dewey-sort | 3518 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-1-4757-3726-4 |
| format | Electronic eBook |
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| id | DE-604.BV042421516 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:09Z |
| institution | BVB |
| isbn | 9781475737264 9781475737288 |
| issn | 1571-5698 |
| language | English |
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| spelling | Shapira, Yair Verfasser aut Matrix-Based Multigrid Theory and Applications by Yair Shapira Boston, MA Springer US 2003 1 Online-Ressource (XVI, 221 p) txt rdacontent c rdamedia cr rdacarrier Numerical Methods and Algorithms 2 1571-5698 Many important problems in applied science and engineering, such as the NavierStokes equations in fluid dynamics, the primitive equations in global climate modeling, the strain-stress equations in mechanics, the neutron diffusion equations in nuclear engineering, and MRIICT medical simulations, involve complicated systems of nonlinear partial differential equations. When discretized, such problems produce extremely large, nonlinear systems of equations, whose numerical solution is prohibitively costly in terms of time and storage. High-performance (parallel) computers and efficient (parallelizable) algorithms are clearly necessary. Three classical approaches to the solution of such systems are: Newton's method, Preconditioned Conjugate Gradients (and related Krylov-space acceleration techniques), and multigrid methods. The first two approaches require the solution of large sparse linear systems at every iteration, which are themselves often solved by multigrid methods. Developing robust and efficient multigrid algorithms is thus of great importance. The original multigrid algorithm was developed for the Poisson equation in a square, discretized by finite differences on a uniform grid. For this model problem, multigrid exhibits extremely rapid convergence, and actually solves the problem in the minimal possible time. The original algorithm uses rediscretization of the partial differential equation (POE) on each grid in the hierarchy of coarse grids that are used. However, this approach would not work for more complicated problems, such as problems on complicated domains and nonuniform grids, problems with variable coefficients, and non symmetric and indefinite equations. In these cases, matrix-based multi grid methods are in order Mathematics Computer science Electronic data processing Computer science / Mathematics Computational Mathematics and Numerical Analysis Numeric Computing Mathematics of Computing Applications of Mathematics Datenverarbeitung Informatik Mathematik Matrizenrechnung (DE-588)4126963-9 gnd rswk-swf Mehrgitterverfahren (DE-588)4038376-3 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Mehrgitterverfahren (DE-588)4038376-3 s 1\p DE-604 Differentialgleichung (DE-588)4012249-9 s 2\p DE-604 Matrizenrechnung (DE-588)4126963-9 s 3\p DE-604 Numerical Methods and Algorithms 2 (DE-604)BV016934978 2 https://doi.org/10.1007/978-1-4757-3726-4 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Shapira, Yair Matrix-Based Multigrid Theory and Applications Numerical Methods and Algorithms Mathematics Computer science Electronic data processing Computer science / Mathematics Computational Mathematics and Numerical Analysis Numeric Computing Mathematics of Computing Applications of Mathematics Datenverarbeitung Informatik Mathematik Matrizenrechnung (DE-588)4126963-9 gnd Mehrgitterverfahren (DE-588)4038376-3 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4126963-9 (DE-588)4038376-3 (DE-588)4012249-9 |
| title | Matrix-Based Multigrid Theory and Applications |
| title_auth | Matrix-Based Multigrid Theory and Applications |
| title_exact_search | Matrix-Based Multigrid Theory and Applications |
| title_full | Matrix-Based Multigrid Theory and Applications by Yair Shapira |
| title_fullStr | Matrix-Based Multigrid Theory and Applications by Yair Shapira |
| title_full_unstemmed | Matrix-Based Multigrid Theory and Applications by Yair Shapira |
| title_short | Matrix-Based Multigrid |
| title_sort | matrix based multigrid theory and applications |
| title_sub | Theory and Applications |
| topic | Mathematics Computer science Electronic data processing Computer science / Mathematics Computational Mathematics and Numerical Analysis Numeric Computing Mathematics of Computing Applications of Mathematics Datenverarbeitung Informatik Mathematik Matrizenrechnung (DE-588)4126963-9 gnd Mehrgitterverfahren (DE-588)4038376-3 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Mathematics Computer science Electronic data processing Computer science / Mathematics Computational Mathematics and Numerical Analysis Numeric Computing Mathematics of Computing Applications of Mathematics Datenverarbeitung Informatik Mathematik Matrizenrechnung Mehrgitterverfahren Differentialgleichung |
| url | https://doi.org/10.1007/978-1-4757-3726-4 |
| volume_link | (DE-604)BV016934978 |
| work_keys_str_mv | AT shapirayair matrixbasedmultigridtheoryandapplications |